Adaptive Conditional Bias-Penalized Kalman Filter for Improved Estimation of Extremes and Its Approximation for Reduced Computation

Abstract

Kalman filter (KF) and its variants and extensions are wildly used for hydrologic prediction in environmental science and engineering. In many data assimilation applications of Kalman filter (KF) and its variants and extensions, accurate estimation of extreme states is often of great importance. When the observations used are uncertain, however, KF suffers from conditional bias (CB) which results in consistent under- and overestimation of extremes in the right and left tails, respectively. Recently, CB-penalized KF, or CBPKF, has been developed to address CB. In this paper, we present an alternative formulation based on variance-inflated KF to reduce computation and algorithmic complexity, and describe adaptive implementation to improve unconditional performance. For theoretical basis and context, we also provide a complete self-contained description of CB-penalized Fisher-like estimation and CBPKF. The results from one-dimensional synthetic experiments for a linear system with varying degrees of nonstationarity show that adaptive CBPKF reduces the root-mean-square error at the extreme tail ends by 20 to 30% over KF while performing comparably to KF in the unconditional sense. The alternative formulation is found to approximate the original formulation very closely while reducing computing time to 1.5 to 3.5 times of that for KF depending on the dimensionality of the problem. Hence, adaptive CBPKF offers a significant addition to the dynamic filtering methods for general application in data assimilation when the accurate estimation of extremes is of importance.

Keywords

• state estimation
• extremes
• conditional bias
• Kalman filter
• adaptive filtering